Abstract
Under suitable conditions, a function f(t) in a principal shift-invariant space can be recovered from its uniform samples { f(n)} n∈ℤ with a simple sampling formula. Provided that the generator ϕ has compact support, we consider the sampling problem in the bigger space V ϕ ≔ {∑a n ϕ(t − n) : a n ∈ ℂℤ}. In this space, there exist infinite functions with the same samples y n = f(n). We show that polynomial growth conditions give the uniqueness: if y n has polynomial growth, there is a unique function of polynomial growth f ∈ V ϕ, satisfying f(n) = y n , n ∈ ℤ. This function is given by the known sampling formula. The same result is proved also when we consider average samples y n = f ∗ h(n).
Acknowledgements
This work has been supported by the grant MTM2009–08345 from the D.G.I. of the Spanish Ministerio de Ciencia y Tecnología.